Just make sure your application is not in sandbox mode and users will be able to use it. Also you might want to enable social discovery so users friends will see they are using the application. Once you get a certain number of users using your application, your application will automatically be found in the search results, but there currently is no longer an application process where Facebook has to approve your application.

Facebook applications are not like iPhone applications. They go live as you create them. You do not have to get them approved. If a user flags your application as inappropriate, Facebook will review your application, and suspend it if it has violations. They will notify you of the action and the reasons, and you can appeal to be re-instated when you have fixed the problem.

You do submit apps to face for approval it your app requires more rights than the minimum. For instance, if you want to read users lists, you will need to get this special right. If you just want to show the users info, you do not have to.

Here is my app from face submission:

Thanks, your submission is pending
You have submitted your app to the Facebook review team. On average, submissions will be reviewed within 7 business days. The following items will be reviewed:

Have you already logged into App Dev section (having joined it as a Developer), and then hit the "Create App" button on top right, created the application-name/hand etc. ? The instructions from there on are self explanatory, s.a. setting shareing attributes etc.

Haven't checked the remaining steps of late, but that's pretty much what it used to be (roughly).